To decrease the costs of operating a lock in a large river, a new

system of operation is proposed. The system will cost $$830,000$ to

design and build. It is estimated that it will have to be reworked every

$10$ years at a cost of $$120,000.$ In addition, an expenditure of $$80,000$

will have to be made at the end of the fifth year for a new type of gear

that will not be available until then. Annual operating costs are

expected to be $$70,000$ for the first $15$ years and $$100,000$ a year

thereafter. Compute the capitalized cost of perpetual service at

$5\mathrm{.}30$ Consider a retired gentleman who starts to collect his Social

Security benefits at the age of $66.$ The monthly check would be close to

$$2,460.$ Assuming that his interest rate is $6\%$ compounded monthly,

answer the following questions:

a$.$ If he lives $20$ years after retirement, what would be the

equivalent total Social Security benefit collected?

b$.$ If he lives $40$ years after retirement, what would be the

equivalent total Social Security benefit collected?

c$.$ Suppose he lives forever. What would be the total amount

collected? Now, comparing this answer with that in $($b$),$ what can

you conclude?